1. Field of the Invention
The present invention relates to a method of determining the twist angle between the rotational axes of each articulation with regard to three articulations constituting the robot wrist, and more particularly to a method of determining kinematic parameters for the robot wrist, in which a method of determining the twist angle between the rotational axes is developed so that the robot wrist can orient arbitrarily at any positions without regard to the location of the robot arm when attaching the robot wrist in which the rotational axes of three articulations do not coincide at one point, to the robot arm of an articulated type.
2. Description of the Prior Art
In general, a robot necessitates six articulations in order for the end-effector of the robot to select a certain position and posture in the three-dimensional workspace. The aforementioned six articulations are composed of three articulations for selecting a position and of three articulations for selecting a posture. Of these, the part having three articulations for selecting a posture is referred to as the wrist. The wrist has to be constructed so that it can orient arbitrarily at that position without regard to the construction of the robot arm, when the aforementioned robot arm equipped with 3 degrees of freedom for determining a posture selected a certain position. Of the robot wrist manufactured up to now, there are two wrists; one is that the rotational axes of each articulation coincide at one point and the other is that the rotational axes of each articulation do not coincide at one point. As for the wrist having the rotational axes of the three articulations which coincide at one point, the fact that the scope in which the robot wrist can be oriented is restricted in accordance with the magnitude of the twist angle between the rotational axes, was publicly disclosed (R. P. Paul, "Kinematics of Robot Wrist, J. of Robotics Research, 1983). In order to eliminate such restriction, a method of determining twist angle between the rotational axes has been proposed. As for the wrist having rotational axes of the three articulations which do not coincide at one point, i.e. having an offset the fact that the scope in which the wrist cannot be oriented exists in accordance with the construction of said three articulations, was disclosed (U.S. Pat. No. 4,823,279, Unimation, Inc., Inventor: William Perzley, etc.). However, a method of determining the twist angle between the rotational axes of three articulations having said rotational axes which do not coincide at one point, has not been disclosed yet. It depends only upon the method of determining the twist angle for robot wrist in which said rotational axes coincide at one point. In consequence, if attaching the wrist having the construction of an offset which kinematic parameters were determined in accordance with the existing method of determining kinematic parameters, to the robot arm of an articulated type, the wrist cannot orient arbitrarily while it was originally intended to do.
The method of determining the twist angle between the rotational axes of three articulations constituting the conventional robot wrist will be explained in more detail as the following.
First, the construction of three articulations constituting the robot wrist is modeled mathematically as shown in FIG. 1 a). In the drawing, as for each articulation, its own coordinate system is established and correlation of front articulation and rear articulation is represented as four each of kinematic parameters. The coordination system established in each articulation is the right-angled system having the constituents of X, Y and Z but FIG. 1 a) shows only the axes of X and Z. Axis (Z.sub.3) is the rotational axis of the first articulation among three articulations constituting the wrist and represents the direction of the robot arm. Axis (Z.sub.4) and axis (Z.sub.5) are rotational axes of the remaining articulations and axis (Z.sub.5) expressly represents the wrist direction. Axis (X.sub.3) means X axis of a coordinate system established in the first articulation and under the initial state which each of the rotational axis was not rotated, axis (X.sub.4) and axis (X.sub.5) of the remaining articulations come to coincide to the X axis of the aforesaid first articulation. Of four kinematic parameters, the most important parameter of determining orienting region of the wrist is the twist angle between Z axes and is denoted by .alpha. in the drawing. Twist angle (.alpha..sub.4) is shown between axis (Z.sub.3) and axis (Z.sub.4), and twist angle (.alpha..sub.5) is shown between axes (Z.sub.4) and (Z.sub.5).
It is known up to now that when twist angle (.alpha..sub.4) and twist angle (.alpha..sub.5) have the different value from each other, an axis (Z.sub.5) cannot exist in a conical space region (cone) (P) formed around axis (Z.sub.3) if axis (Z.sub.5) rotates about axis (Z.sub.4) as shown in FIG. 1 b) (R. P. Paul, "Kinematics of Robot Wrist", J. of Robotics Research, 1983). This means that when axis (Z.sub.3) is the direction of the robot arm and axis (Z.sub.5) is the wrist direction, there exists a certain space between axis (Z.sub.3) and axis (Z.sub.5) and thus the robot wrist cannot orient in this region. However, if twist angle (.alpha..sub.4) and twist angle (.alpha..sub.5) have the equal value from each other, the direction of the robot arm may coincide to the wrist direction and thus the wrist can orient arbitrarily in the region indicated with a slant line. Also, the region (R) which is not represented as a slant line in FIG. 1 c) completely disappears if twist angle (.alpha..sub.4) and twist angle (.alpha..sub.5) become 90.degree. respectively.
As the result, the technique of determining kinematic parameters for the robot wrist used up to date requires that twist angle (.alpha..sub.4) and twist angle (.alpha..sub.5) should have equal value and that each of the angle size becomes 90.degree. respectively. As for most conventional robot wrist, such method of determining the parameters was used.
Meanwhile, the aforesaid method of determining kinematic parameters applies to the wrist with an offset having the construction in which the rotational axes do not coincide at one point. Especially if attaching the wrist having such construction, to the robot arm of an articulated type, original object thereof cannot be achieved.
Namely, as for such kind of robot, there exist some region in which the robot wrist cannot be oriented arbitrarily. Actually in the industrial fields, such kind of robot is used mostly but all of them have the drawbacks mentioned above.
The problems of the robot being used at present will be explained in detail with reference to the accompanying drawings. FIG. 2 shows a mathematically modeled robot with wrist in which the rotational axes of three articulations do not coincide at one point, which was attached to the arm of an articulated type.
Kinematic parameters of said robot are shown in the following Table 1.
TABLE 1 ______________________________________ Articulation .alpha. a d .theta. ______________________________________ 1 90 0 0 .theta..sub.1 2 0 l.sub.2 0 .theta..sub.2 3' 0 l.sub.3 0 .theta..sub.3 3 -90 0 0 -90 4 90 0 0 .theta..sub.4 5 -90 0 d.sub.5 .theta..sub.5 6 0 0 0 .theta..sub.6 ______________________________________
In the above Table 1, twist angle (.alpha..sub.4) and twist angle (.alpha..sub.5) of articulation (4) and articulation (5) are 90.degree. respectively. These values are those determined according to the conventional method of determining kinematic parameters. The denotation (-) represents the case where the initial coordinate position is the opposite direction. With regard to a distance (d) between the rotational axes, it can be seen from the above Table that there exist a distance (d.sub.5) between articulation (4) and articulation (5). The value of this distance (d.sub.5) becomes 0 if the rotational axes of three articulations coincide at one point.
If the distance (d.sub.5) is 0, no problem arises in the conventional method of determining kinematic parameters. If it is not 0 as mentioned above, problems arise as the following.
As shown in FIG. 3, supposing that the robot should assume a posture of a at point (P.sub.5) in workspace, the robot should determine the posture of a positioning at point (P.sub.5) and by moving articulation (4) and articulation (5). However, axis (Z.sub.5) representing the wrist direction moves only outside the conical region (P.sup.o) and cannot reach inside the conical region (P.sup.o) as shown in FIG. 4, even if said robot causes articulation (4) and articulation (5) to rotate. That is, the robot wrist cannot assume the posture of a in case there exist a virtual posture a within the aforesaid conical region (P.sup.o).
In conclusion, in case of the robot wrist having the rotational axes of three articulations which do not coincide at one point, there exist some region in which the robot wrist cannot be oriented at a virtual position in workspace even though twist angle (.alpha..sub.4) and twist angle (.alpha..sub.5) are equal and 90.degree. according to the conventional method of determining kinematic parameters thereof. As the result, kinematic parameters, i.e. twist angles (.alpha..sub.4) and (.alpha..sub.5) should not be equal or 90.degree. when attaching the wrist with an offset, to the robot arm of an articulated type.